# University of Chicago - Geometric Analysis Seminar

#### Department of Mathematics

We welcome all those who are interested to join us on Tuesdays, 1:00–2:00 PM, on zoom. To receive the zoom information please write an email to Daniel or Henrik.

### Winter 2021

#### January 19, 2021 Tue, 1:00–2:00PM

Hans-Joachim Hein, Münster

Smooth asymptotics for collapsing Calabi-Yau metrics : Yau's solution of the Calabi conjecture provided the first nontrivial examples of Ricci-flat Riemannian metrics on compact manifolds. Attempts to understand the degeneration patterns of these metrics in families have revealed many remarkable phenomena over the years. I will review some aspects of this story and present recent joint work with Valentino Tosatti where we obtain a complete asymptotic expansion (locally uniformly away from the singular fibers) of Calabi-Yau metrics collapsing along a holomorphic fibration of a fixed compact Calabi-Yau manifold. This relies on a new analytic method where each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.

#### January 26, 2021 Tue, 1:00–2:00PM

Fritz Hiesmayr, UCL

An Urbano-type theorem for the Allen-Cahn equation : The Allen-Cahn equation is a semilinear elliptic PDE modelling phase transitions in two-phase media. In recent years this has found applications in geometry due to its link with minimal hypersurfaces. We present an analogue of Urbano's theorem about minimal surfaces in the round three-sphere. Our result is a rigidity theorem for solutions of the Allen-Cahn equation in S^3 with small index. They are symmetric, and vanish either on a minimal sphere or a Clifford torus. One key observation is that the nodal sets of two distinct solutions must have non-empty intersection.

#### February 2, 2021 Tue, 1:00–2:00PM

Daren Cheng, Waterloo

TBA : TBA

#### February 9, 2021 Tue, 1:00–2:00PM

Aleksander Doan, Columbia

TBA : TBA

#### February 16, 2021 Tue, 1:00–2:00PM

Costante Bellettini, UCL

TBA : TBA

#### February 23, 2021 Tue, 1:00–2:00PM

Robin Neumayer, Northwestern

TBA : TBA

### Autumn 2020

#### October 6, 2020 Tue, 11:00am–12pm

Dmitry Jakobson, McGill

Zero and negative eigenvalues of conformally covariant operators, and nodal sets in conformal geometry : We first describe conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators (such as Yamabe or Paneitz operator). We discuss applications to curvature prescription problems. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension greater than 2. We show that 0 is generically not an eigenvalue of the conformal Laplacian. If time permits, we shall discuss related results on manifolds with boundary, as well as for weighted graphs. This is joint work with Y. Canzani, R. Gover, R. Ponge, A. Hassannezhad, M. Levitin, M. Karpukhin, G. Cox and Y. Sire.

#### October 13, 2020 Tue, 11:00am–12pm

Christos Mantoulidis, Brown

Ancient mean curvature flows, gradient flows, and Morse index : This talk gives an overview of two recent joint works. The first work, joint with Kyeongsu Choi, studies closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds. Our methods classify ancient solutions coming out of a critical point, with very mild decay assumptions, establishing that they are parametrized by unstable eigenfunctions of the critical point. Applied to mean curvature flow, these methods imply an arbitrary dimension and codimension classification of ancient mean curvature flows of closed submanifolds of S^n with low area. The second work, joint with Otis Chodosh, Kyeongsu Choi, and Felix Schulze, extends these methods to study ancient mean curvature flows lying on one side of asymptotically conical shrinking solitons. As an application, we partially confirm the conjectured non-linear instability of asymptotically conical singularity models in mean curvature flow.

#### October 20, 2020 Tue, 5:00pm–6:00pm

Jonathan Zhu, Princeton

Explicit Łojasiewicz inequalities for mean curvature flow shrinkers : Łojasiewicz inequalities are a popular tool for studying the stability of geometric structures. For mean curvature flow, Schulze used Simon’s reduction to the classical Łojasiewicz inequality to study compact tangent flows. Colding and Minicozzi instead used a direct method to prove Łojasiewicz inequalities for round cylinders. We’ll discuss similarly explicit Łojasiewicz inequalities and applications for Clifford shrinkers and certain other cylinders.

#### October 27, 2020 Tue, 11:00am–12:00pm

Robert Haslhofer, University of Toronto

Ancient flows and the mean-convex neighborhood conjecture : I will explain our recent proof of the mean-convex neighborhood conjecture. The key is a classification result for ancient asymptotically cylindrical mean curvature flows. The 2-dimensional case is joint work with Choi and Hershkovits, and the higher-dimensional case is joint with Choi, Hershkovits and White.

#### November 3, 2020 Tue, 11:00am–12:00pm

G2-monopoles (a summary) : This talk is aimed at reviewing what is known about G2-monopoles and motivate their study. After this, I will mention some recent results obtained in collaboration with Ákos Nagy and Daniel Fadel which investigate the asymptotic behavior of G2-monopoles. Time permitting, I will mention a few possible future directions regarding the use of monopoles in G2-geometry.

#### November 10, 2020 Tue, 11:00am–12:00pm

Ao Sun, University of Chicago

Generic Mean Curvature Flow and Generalizations of Entropy. : Mean curvature flow entropy was introduced by Colding-Minicozzi, and it is a very important quantity in the study of mean curvature flow and related geometric problem. In particular, mean curvature flow entropy plays a crucial role in the study of generic mean curvature flow. I will discuss two generalizations of mean curvature flow entropy: one is a localized version of entropy, another one is entropy in a closed manifold. I will discuss how to use these generalizations of entropy to rule out some pathological asymptotic behaviors of mean curvature flow.

#### November 17, 2020 Tue, 11:00am–12:00pm

Liam Mazurowski, University of Chicago

CMC doublings of minimal surfaces via min-max : An interesting problem in differential geometry is to try to understand the space of constant mean curvature surfaces (CMCs) in a given manifold. Recently Zhou and Zhu developed a min-max theory for constructing CMCs, and used it to show that any manifold M of dimension between 3 and 7 contains a smooth, almost-embedded CMC hypersurface of mean curvature h for every h > 0. In this talk, I will explain how this min-max theory can be used to construct CMC doublings of certain minimal surfaces in 3-manifolds. Such CMC doublings were previously constructed for minimal hypersurfaces in M^n with n > 3 by Pacard and Sun using gluing methods.

#### November 24, 2020 Tue, 11:00am–12:00pm

Shubham Dwivedi, Humboldt University of Berlin

Deformation theory of nearly G_2 manifolds : We will discuss the deformation theory of nearly G_2 manifolds. After defining nearly G_2 manifolds, we will describe some identities for 2 and 3 forms on such manifolds. We will introduce a Dirac type operator which will be used to prove new results on the cohomology of nearly G_2 manifolds. Along the way we will reprove a result of Alexandrov—Semmelman on the space of infinitesimal deformation of nearly G_2 structures. Finally, we will prove that the infinitesimal deformations of the homogeneous nearly G_2 structure on the Aloff--Wallach space are obstructed to second order. The talk is based on a joint work with Ragini Singhal (University of Waterloo).

#### December 1, 2020 Tue, 11:00am–12:00pm

Alexandre Girouard, Universite de Laval

Optimal isoperimetric upper bounds for Steklov eigenvalues of planar domains : Given a compact Riemannian manifold M, the eigenvalues of the Dirichlet-to-Neumann map acting on the boundary of M are known as Steklov eigenvalues of M. I will present a complete solution of the isoperimetric problem for each perimeter-normalized Steklov eigenvalue of planar domains: the best upper bound for the $k+1$-th perimeter-normalized Steklov eigenvalue is $8k\pi$. The proofs are based on continuity properties for variational eigenvalues associated to Radon measures on compact manifolds. In particular, starting with any domain in the plane, we constructed a sequence of subdomains with its $k+1$-th perimeter-normalized Steklov eigenvalue converging to $8k\pi$. These subdomains are obtained by removing small disks from the initial domain, in the spirit of homogenization theory. Properties of maximizing sequences will also be discussed, showing in particular that any sequence of domains with prescribed perimeter with its first nonzero Steklov eigenvalue converging to $8\pi$ must collapse to a point. This talk is based on recent papers with with Antoine Henrot (U. de Lorraine), Mikhail Karpukhin (Caltech) and Jean Lagacé (UCL).

### Winter 2020

#### January 14, 2020 Tue, 3:30–4:30PM, Eckhart 202

Dan Lee, CUNY

Progress on Bartnik’s stationary conjecture : Given a compact initial data set with boundary, the Bartnik problem is the problem of finding asymptotically flat initial data that extends given data in such a way that it minimizes mass among all possible (admissible) extensions satisfying the dominant energy condition. Bartnik conjectured that if such a minimizer exists, then it must be stationary in the extended region. In the time-symmetric case, this was settled by Corvino (in which case the stationary extension is actually static). In joint work with Lan-Hsuan Huang of the University of Connecticut, we are able to prove in the general case that a minimizing extension must be vacuum stationary outside some large compact set. Our proof involves finding ways to locally deform the geometry of an initial data set in such a way that we have useful control over the energy-momentum density.

#### January 21, 2020 Tue, 3:30–4:30PM, Eckhart 202

Gigliola Staffilani, MIT

Some results on the almost everywhere convergence of the Schrodinger flow : In this work we are concerned with the question of almost everywhere convergence of the nonlinear Schrodinger flow as time tends to zero, both in the continuous and the periodic case. We will review the extraordinary progress made in the linear continuous case and we will illustrate some progress recently made in the nonlinear case using both a deterministic and a probabilistic approach. This is joint work with E. Compaan and R. Luca.

#### January 21, 2020 Tue, 4:30–5:30PM, Ryerson 276

Isoperimetric inequality in spaces of nonpositive curvature : The classical isoperimetric inequality states that in Euclidean space spheres provide unique enclosures of least perimeter for any given volume. In this talk we show that this inequality generalizes to spaces of nonpositive curvature, or Cartan-Hadamard manifolds, as conjectured by Aubin, Gromov, Burago, and Zalgaller in 1970s and 80s. The proof is based on a comparison formula for total curvature of level sets in Riemannian manifolds. This is joint work with Joel Spruck.

#### January 31, 2020 Fri, 2:00–3:00PM, Eckhart 207

Philippe G. LeFloch, Sorbonne University and CNRS

Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions : I will present recent progress on the global evolution problem for self-gravitating matter. (1) For Einstein's constraint equations, motivated by a scheme proposed by Carlotto and Schoen I will show the existence of asymptotically Euclidean Einstein spaces with low decay; joint work with T. Nguyen. (2) For Einstein's evolution equations in the regime near Minkowski spacetime, I will show the global nonlinear stability of massive matter fields; joint work with Y. Ma. (3) For the colliding gravitational wave problem, I will show the existence of weakly regular spacetimes containing geometric singularities across which junction conditions are imposed; joint work with B. Le Floch and G. Veneziano.

#### February 4, 2020 Tue, 3:30–4:30PM, Eckhart 202

Pedro Gaspar, UChicago

Solutions of the Allen-Cahn equation on closed manifolds in the presence of symmetry : In recent years, there has been remarkable development about the connection between the Allen-Cahn equation and minimal hypersurfaces. A result due to F. Pacard and M. Ritoré shows that given a minimal hypersurface Γ in a closed manifold M, we can find solutions of this PDE whose nodal sets approximate Γ, provided it is a nondegenerate critical point of the area functional. This condition is often too restrictive, as many interesting examples of ambient manifolds have continuous groups of symmetries, and the corresponding variations preserve the area of any hypersurface. In this talk, we explain how to obtain a similar existence result in the case where all Jacobi fields of Γ arise from ambient isometries. We also derive some geometric consequences, and describe a second order convergence result for multiplicity-one solutions, in terms of the corresponding linearized operator and its eigenvalues. This is joint work with R. Caju.

#### February 11, 2020 Tue, 3:30–4:30PM, Eckhart 202

Celso dos Santos Viana, UC Irvine

Isoperimetry and volume preserving stability in real projective spaces : In this talk I will address the problem of classifying volume preserving stable constant mean curvature hypersurfaces in Riemannian manifolds. I will present recent classification in the real projective space of any dimension and, consequently, the solution of the isoperimetric problem.

#### February 18, 2020 Tue, 3:30–4:30PM, Eckhart 202

Antoine Song, UC Berkeley

Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces : We will explain how, for minimal hypersurfaces with uniformly bounded area, the topology and the singular set can be controlled by the Morse index. We use this to study the sequence of minimal surfaces constructed by min-max theory in a given closed 3-manifold. Some natural open questions will be introduced.

#### February 25, 2020 Tue, 3:30–4:30PM, Eckhart 202

Peter McGrath, UPenn

Generalizing the Linearized Doubling Approach and New Minimal Surfaces and Self-Shrinkers via Doubling. : T I will discuss recent work (with N. Kapouleas) on generalizing the Linearized Doubling approach to apply (under reasonable assumptions) to doubling arbitrary closed minimal surfaces in arbitrary Riemannian three-manifolds without any symmetry requirements. More precisely, given a family of LD solutions on a closed minimal surfaces embedded in a Riemannian three-manifold, where an LD solution is a solution of the Jacobi equation with logarithmic singularities, we prove a general theorem which states that if the family satisfies certain conditions, then a new minimal surface can be constructed via doubling, with catenoidal bridges replacing the singularities of one of the LD solutions. The construction of the required LD solutions is currently only understood when the surface and ambient manifold possess O(2) symmetry and the number of bridges is chosen large enough along O(2) orbits. In this spirit, we use the theorem to construct new self-shrinkers of the mean curvature flow via doubling the spherical self-shrinker and new complete embedded minimal surfaces of finite total curvature in the Euclidean three-space via doubling the catenoid.

#### March 10, 2020 Tue, 3:30–4:30PM, Eckhart 202

Ruobing Zhang, Stony Brook

TBA : TBA

### AUTUMN 2019

#### October 8, 2019 Tue, 3:30–4:30PM, Eckhart 202

Antonio De Rosa, Courant Institute, NYU

Elliptic integrands in geometric analysis : We present the recent tools we developed to prove existence and regularity properties of the critical points of anisotropic functionals. In particular, we present our extension of Allard's celebrated rectifiability theorem to the setting of varifolds with bounded anisotropic first variation. We apply this result to solve the set-theoretic anisotropic Plateau problem. We obtain as corollaries easy solutions to three different formulations of the Plateau problem, introduced by Reifenberg, by Harrison-Pugh and by David. Moreover, we prove an anisotropic counterpart of Allard's compactness theorem for integral varifolds. To conclude, we focus on the anisotropic isoperimetric problem. We prove the anisotropic counterpart of Alexandrov's characterization of volume-constrained critical points among finite perimeter sets. Furthermore we derive stability inequalities associated to this rigidity theorem. Some of the presented theorems are joint works with De Lellis, De Philippis, Ghiraldin, Gioffre, Kolasinski and Santilli.

#### October 15, 2019 Tue, 3:30–4:30PM, Eckhart 202

Daren Cheng, UChicago

Bubble tree convergence of conformally cross product preserving maps : We study a class of weakly conformal maps, called Smith maps, which parametrize associative 3-folds in 7-manifolds equipped with G2-structures. These maps satisfy a system of first-order PDEs generalizing the Cauchy-Riemann equation for J-holomorphic curves, and we are interested in their bubbling phenomena. Specifically, we first prove an epsilon-regularity theorem for Smith maps in W^{1, 3}, and then explain how that combines with conformal invariance to yield bubble trees of Smith maps from sequences of such maps with uniformly bounded 3-energy. When the G2-structure is closed, we show that both the 3-energy and the homotopy are preserved in the bubble tree limit. The result can be viewed as an associative analogue of the bubble tree convergence theorem for J-holomorphic curves. This is joint work with Spiro Karigiannis and Jesse Madnick.

#### October 22, 2019 Tue, 3:30–4:30PM, Eckhart 202

Zheng Huang, CUNY-Staten Island

Bifurcation for closed minimal surfaces in hyperbolic three-manifolds : In the 1970s, Uhlenbeck initiated a program to study questions about existence, multiplicity of closed minimal surfaces in hyperbolic three-manifolds as well as applications in various related areas. In this talk, based on joint work with M. Lucia and G. Tarantello, I will describe several results in these directions.

#### October 29, 2019 Tue, 3:30–4:30PM, Eckhart 202

Xiaodong Wang, Michigan State

On the size of compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary : Given a compact Riemannian manifold with nonnegative Ricci curvature and convex boundary it is interesting to estimate its size in terms of the volume, the area of its boundary etc. I will discuss some open problems and present some partial results.

#### November 5, 2019 Tue, 3:30–4:30PM, Eckhart 202

Mikhail Karpukhin, UC Irvine

Isoperimetric inequalities for Laplacian eigenvalues: recent developments : The Laplacian is a canonical second order elliptic operator defined on any Riemannian manifold. The study of upper bounds for its eigenvalues under the volume constraint is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. The particular interest to this problem stems from a surprising connection to the theory of minimal surfaces in spheres. In the present talk we will survey some recent results in the area with an emphasis on the role played by the index of minimal surfaces. In particular, we will discuss some recent applications, including a new lower bounds for the index of minimal spheres as well as the optimal isoperimetric inequality for Laplacian eigenvalues on the projective plane.

#### November 12, 2019 Tue, 3:30–4:30PM, Eckhart 202

Tristan Collins, MIT

Some results in Strominger-Yau-Zaslow Mirror Symmetry : Mirror symmetry originally arose as a mysterious duality between Calabi-Yau threefolds, interchanging complex and symplectic structures. This duality has since expanded to include a much broader collection of objects, including Fano manifolds, and Landau-Ginzburg models. Two fundamental themes in mirror symmetry are (1) the existence of special Lagrangian fibrations, as conjectured by Strominger-Yau-Zaslow and (2) the correspondence between “stable” objects as predicted in work of Thomas-Yau, and Douglas. Here, stable objects are meant to be special Lagrangian manifolds on the symplectic side, and holomorphic bundles with canonical metrics, on the complex side. I will report on recent results in both of these directions. This talk with discuss joint works with A. Jacob, Y.-S. Lin, and S.-T. Yau.

#### November 19, 2019 Tue, 3:30–4:30PM, Eckhart 202

Yang Li, IAS

Taub-NUT and Ooguri-Vafa: from 2D to 3D : Taub-NUT and Ooguri-Vafa metrics are S^1 invariant Calabi-Yau metrics in complex dimension 2 constructed via the Gibbons-Hawking ansatz. They feature prominently in the complex 2D case of the SYZ conjecture. After reviewing the basics we explain how a number of first principles dictate their construction. This insight enables us to generalize the construction to complex dimension 3, which is expected to be relevant for the SYZ conjecture.

#### November 26, 2019 Tue, 3:30–4:30PM, Eckhart 202

Romain Petrides, IMJ, Paris Diderot University

Critical metrics for Laplace eigenvalues on Riemannian surfaces : We investigate the general link between the critical unit area metrics for eigenvalues of the Laplace operator on closed surfaces, and minimal immersions of these surfaces by eigenfunctions. We will discuss the existence or non existence of such objects by variational methods.

#### December 3, 2019 Tue, 3:30–4:30PM, Eckhart 202

Demetre Kazaras, Stony Brook

Desingularizing 4-manifolds with positive scalar curvature : We study 4-manifolds of positive scalar curvature (psc) with severe metric singularities along points and embedded circles, establishing a desingularization process. To carry this out, we show that the bordism group of closed 3-manifolds with psc metrics is trivial, using scalar-flat K{\"a}hler ALE surfaces recently discovered by Lock-Viaclovsky. This allows us to prove a non-existence result for singular psc metrics on enlargeable 4-manifolds, partially confirming a conjecture of Schoen. We will also mention a new lower bound for the mass of 3d asymptotically flat manifolds with nonnegative scalar curvature in joint work with Bray, Khuri, and Stern.

### SPRING 2019

#### April 16, 2019 Tue, 3:00–4:00PM, Eckhart 202

Lucas Ambrozio, IAS

Comparing the total volume of a closed Riemannian manifold and the geometric invariants related to its minimal hypersurfaces : The variational methods used to find closed embedded minimal hypersurfaces in a closed Riemannian manifold allow one to define several notions of "systole" and "width", which can be regarded as functionals on the space of Riemannian metrics. In this talk, we will be interested in the properties of some of these functionals on the space of unit volume metrics, focusing on the case of three dimensional spheres and real projective spaces. In particular, we will look for (sharp) upper bounds on certain subsets of metrics and describe necessary conditions satisfied by local maxima. This is joint work with Rafael Montezuma.

#### April 23, 2019 Tue, 3:00–4:00PM, Eckhart 202

Nicolau Aiex, UBC

TBA : TBA

#### April 23, 2019 Tue, 4:00–5:00PM, Eckhart 202

Davi Maximo, UPenn

On the topology and index of minimal surfaces : For an immersed minimal surface in R^3, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resolves several conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show that there are no complete two-sided immersed minimal surfaces in R^3 of index two, complete embedded minimal surface with index three, or complete one-sided minimal immersion with index one. This is joint work with Otis Chodosh.

#### April 30, 2019 Tue, 3:00–4:00PM, Eckhart 202

Alessandro Carlotto, IAS/ETH-Zurich

Constrained deformations of positive scalar curvature metrics : I will present a series of results concerning the interplay between two different curvature conditions, in the special case when these are given by pointwise inequalities on the scalar curvature of a manifold, and the mean curvature of its boundary. Such results lie at two conceptual levels: on the one hand at the level of compatibility (i.e. is it possible to simultaneously satisfy the bounds, and what are the resulting topological implications), on the other hand at the level of moduli space structure (i.e. what can one say about the homotopy type of the associated space of metrics, when not empty, quotiented by the diffeomorphism group of the background manifold). In particular, we give a complete topological characterization of those compact 3-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary and, in any such case, we prove that the associated moduli space of metrics is path-connected. Furthermore, we show how our methods can be refined so to construct continuous paths of positive scalar curvature metrics with minimal boundary, and to obtain analogous conclusions in that context as well. Our work relies on a combination of earlier fundamental contributions by Gromov-Lawson and Schoen-Yau, on the smoothing procedure designed by Miao, and on the interplay of Perelman's Ricci flow with surgery and conformal deformation techniques introduced by Codá Marques in dealing with the closed case. This lecture is based on joint work with Chao Li (Princeton University).

#### May 7, 2019 Tue, 3:00–4:00PM, Eckhart 202

Harrison Pugh, Notre Dame

TBA : TBA

#### May 28, 2019 Tue, 3:00–4:00PM, Eckhart 202

Ben Sharp, University of Leeds

Global estimates for harmonic maps from surfaces : A celebrated theorem of F. Hélein guarantees that a weakly harmonic map from a two-dimensional domain is always smooth. The proof is of a local nature and assumes that the Dirichlet energy is sufficiently small; under this condition it is possible to re-write the harmonic map equation using a suitably chosen frame which uncovers non-linearities with more favourable regularity properties (so-called div-curl or Wente structures). We will prove a global estimate for harmonic maps without assuming a small energy bound, utilising a powerful theory introduced by T. Rivière. Along the way we will highlight the relevance of Wente-type estimates in neighbouring areas of geometric analysis, and hint as to why the analogous higher-dimensional global estimate remains a challenging open problem. This is a joint work with Tobias Lamm.

### WINTER 2019

#### January 15, 2019 Tue, 3:00–4:00PM, Eckhart 202

Henrik Matthiesen, The University of Chicago

The systole of large genus minimal surfaces in positive Ricci curvature : We prove that the systole (or more generally, any $k$-th homology systole) of a minimal surface in an ambient three manifold of positive Ricci curvature tends to zero as the genus of the minimal surfaces becomes unbounded. This is joint work with Anna Siffert.

#### January 24, 2019 Thursday, 2:00–3:00PM, Eckhart 202

Xin Zhou, UCSB

Multiplicity One Conjecture in Min-max theory : I will present a recent proof of the Multiplicity One Conjecture in Min-max theory. This conjecture was raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one. As direct corollaries, it implies the generalized Yau's conjecture for such manifolds with positive Ricci curvature, which says that there exist infinitely many pairwise non-isometric minimal hypersurfaces, and the Weighted Morse Index Bound Conjecture by Marques and Neves.

#### January 30, 2019 Wed, 3:00–4:00PM, Eckhart 202

Tristan Riviere, ETH

The Cost of the Sphere Eversion and the 16\pi Conjecture : How much does it cost...to knot a closed simple curve ? To cover the sphere twice ? to realize such or such homotopy class ? ...etc. All these questions consisting of assigning a "canonical" number and possibly an optimal "shape" to a given topological operation are known to be mathematically very rich and to bring together notions and techniques from topology, geometry and analysis. In this talk we will concentrate on the operation consisting of turning inside out the 2 sphere in the 3 dimensional space. Since Smale's proof in 1959 of the existence of such an operation the search for effective realizations of such eversions has triggered a lot of fascination and works in the math community. The absence in nature of matter that can interpenetrate and the quasi impossibility, up to the advent of virtual imaging, to experience this deformation is maybe the reason for the difficulty to develop an intuitive approach on the problem. We will present the optimization of Sophie Germain conformally invariant elastic energy for the eversion. Our efforts will finally bring us to consider more closely an integer number together with a mysterious minimal surface.

#### Feb 12, 2019 Tue, 3:00–4:00PM, Eckhart 202

Vanderson Lima, UFRGS

Stable Minimal Surfaces and the topology of 3-Manifolds : Meeks, Perez and Ros conjectured that a closed Riemannian 3-manifold which does not contain any closed embedded stable minimal surface must be diffeomorphic to a quotient of the 3-sphere. In this talk we show a counter example for this conjecture. We also discuss how to correct the conjecture so that it holds true.

#### Feb 19, 2019 Tue, 3:00–4:00PM, Eckhart 202

Casey Kelleher, Princeton University

INDEX-ENERGY ESTIMATES FOR YANG-MILLS CONNECTIONS AND EINSTEIN METRICS : We prove a conformally invariant estimate for the index of Schrodinger operators acting on vector bundles over four-manifolds, related to the classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we obtain a bound for the index in terms of its energy which is conformally invariant, and captures the sharp growth rate. Furthermore we derive an index estimate for Einstein metrics in terms of the topology and the Einstein-Hilbert energy. Lastly we derive conformally invariant estimates for the Betti numbers of an oriented four-manifold with positive scalar curvature.

#### Feb 26, 2019 Tue, 3:00–4:00PM, Eckhart 202

Giuseppe Tinaglia, King's College

The geometry of constant mean curvature surfaces in Euclidean space : In this talk I will begin by reviewing classical geometric properties of constant mean curvature surfaces, H>0, in R^3. I will then talk about several more recent results for surfaces embedded in R^3 with constant mean curvature, such as curvature and radius estimates for simply-connected surfaces embedded in R^3 with constant mean curvature. Finally I will show applications of such estimates including a characterisation of the round sphere as the only simply-connected surface embedded in R^3 with constant mean curvature and area estimates for compact surfaces embedded in a flat torus with constant mean curvature and finite genus. This is joint work with Meeks.

#### Feb 26, 2019 Tue, 3:00–4:00PM, Eckhart 202

Gregory Chambers, Rice University

Geometric stability of the Coulomb energy : Suppose that $A$ is a measurable subset of $\mathbb{R}^3$ of finite measure. The Coulomb energy of $A$ is the double integral over $A$ of $1/|x-y|$, and is maximized when $A$ is a ball. If the Coulomb energy is close to maximal, then is $A$ geometrically close to a ball? We will answer this question, and will compare it to the quantitative isoperimetric inequality. We will also discuss the analogous situation in higher dimensions. This is joint work with Almut Burchard

#### Mar 5, 2019 Tue, 3:00–4:00PM, Eckhart 202

Yaiza Canzani, UNC

Understanding the growth of Laplace eigenfunctions : In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Using the description of concentration, we obtain quantitative improvements on the known bounds in a wide variety of settings.

#### Mar 12, 2019 Tue, 3:00–4:00PM, Eckhart 202

Yu Wang, Northwestern University

Sharp estimate of global Coulomb gauge in dimension 4 : Consider a principle SU(2)-bundle P over a compact 4-manifold M and a W^{1,2}-connection A of P satisfying \|F_A\|_{L^2(M)}\le \Lambda. Our main result is the existence of a global section \sigma: M\to P with controllably many singularities such that the connection form \sigma^*A satisfies the Coulomb equation d^*(\sigma^*A)=0 and moreover admits a sharp estimate. In this talk, we first recall some preliminaries and then outlines the proof. If time allows we shall elaborate on the ideas to overcome the main difficulties in this problem, which include an epsilon-regularity theorem for the Coulomb gauge, an annular-bubble regions decomposition for the curvature, and studying the singularity behavior of the Coulomb gauge on each annular and bubble region.

#### Mar 19, 2019 Tue, 3:00–4:00PM, Eckhart 202

Rayssa Caju,

ASYMPTOTIC BEHAVIOUR OF SOLUTIONS FOR A COUPLED ELLIPTIC SYSTEM IN THE PUNCTURED BALL : Our main goal is to study the asymptotic behavior near an isolated singularity of local solutions for certain strongly coupled critical elliptic systems that, from the viewpoint of conformal geometry, are pure extensions of Yamabe-type equations. There has been considerable interest in recent years in proving compactness results for this type of systems since such type of problems provides a natural background for the interplay between geometry and asymptotic analysis. We prove a sharp result on the removability of the isolated singularity for all components of the solutions when the dimension is less than or equal to five and minus the potential of the operator in the system is cooperative.

### SPRING 2018

#### April 3, 2018 Tue, 3:45–4:45PM, Eckhart 207

Lucas Ambrosio, University of Warwick

Sequences of minimal surfaces with bounded index in three-manifolds : We explore a few consequences of the bubbling analysis developed by Buzano and Sharp, giving a detailed description of how a sequence of closed embedded minimal surfaces of bounded index in a three-manifold degenerates as we pass to a "converging" subsequence. In particular, a few new compactness result are obtained. This is a joint work with Reto Buzano (QMUL), A. Carlotto (ETH) and B. Sharp (Leeds).

#### April 10, 2018 Tue, 3:45–4:45PM, Eckhart 207

Costante Bellettini, University College of London

Stable constant-mean-curvature hypersurfaces: regularity and compactness. : This talk describes a recent joint work of the speaker with N. Wickramasekera (Cambridge). The work develops a regularity theory, with an associated compactness theorem, for weakly defined hypersurfaces (codimension 1 integral varifolds) of a smooth Riemannian manifold that are stationary and stable on their regular parts for volume preserving ambient deformations. The main regularity theorem gives two structural conditions on such a hypersurface that imply that, away from a set of codimension 7 or higher, the hypersurface is locally either a single smoothly embedded disk or precisely two smoothly embedded disks intersecting tangentially. Easy examples show that neither structural hypothesis can be relaxed. An important special case is when the varifold corresponds to the boundary of a Caccioppoli set, in which case the structural conditions can be considerably weakened. An "effective version" of the compactness theorem has been (a posteriori) established in collaboration with O. Chodosh and N. Wickramasekera.

#### April 17, 2018 Tue, 3:00–4:00PM, Eckhart 202

Daniel Agress, University of California Irvine

Existence results for the nonlinear Hodge minimal surface energy. : The nonlinear Hodge minimal surface energy, first studied by Sibner and Sibner in the 1970's, has applications to minimal surfaces, bounded variation functions, and the Born Infeld theory of electromagnetism. In this talk, we will prove an existence and nonexistence result for minimizers of the energy. In particular, we show that for a compact Riemannian manifold and cohomology class $[\alpha] \in H^k(M)$, minimizers always exist when $k=1$, but counterexamples exist when $k>1$. We will also describe the how the energy can be viewed as a regularization of the BV energy.

#### April 24, 2018 Tue, 3:45–4:45PM, Eckhart 207

Mikhail Karpukhin, McGill University

Laplace eigenvalues and minimal surfaces in spheres : We will give an overview of some recent estimates for Laplace eigenvalues on Riemannian surfaces. In particular, we will discuss the connection of optimal isoperimetric inequalities with minimal surfaces and harmonic maps. Finally, this connection will be used in order to prove the sharp upper bound for all Laplace eigenvalues on the two-dimensional sphere. The talk is based on a joint work with N. Nadirashvili, A. Penskoi and I. Polterovich.

#### April 30, 2018 Mon, 3:45–4:45PM, Eckhart 207 Please note the unusual day

Peter Smillie, Harvard University

Entire spacelike surfaces of constant curvature in Minkowski 3-space : We prove that every regular domain in Minkowski 3-space which is not a wedge contains a unique entire spacelike surface with constant intrinsic curvature equal to -1. This completes the classification of such surfaces in terms of their domains of dependence, for which partial results were obtained by Li, Guan-Jian-Schoen, and Bonsante-Seppi. Using this result, we obtain an analogous classification of entire spacelike surfaces with constant mean curvature (CMC). We'll apply these ideas to the Minkowski problem of prescribed curvature and to the construction CMC times in 2+1 relativity, and we'll see what we can say about the problem of deciding when the induced hyperbolic metric on an entire surface is complete. Everything is joint with Francesco Bonsante and Andrea Seppi.

#### May 8, 2018 Tue, 3:45–4:45PM, Eckhart 207

Rafael Montezuma, Princeton University

A mountain pass theorem for minimal hypersurfaces with fixed boundary : In this talk, we will be concerned with the existence of a third embedded minimal hypersurface, of mountain-pass type, spanning a closed submanifold B contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound B. In order to do so, we develop min-max methods similar to those of the recent work of De Lellis and Ramic, adapted to the discrete setting of Almgren and Pitts.

#### May 15, 2018 Tue, 3:45–4:45PM, Eckhart 207

Pei-Ken Hung, Columbia University

The smoothing time of convex inverse mean curvature flows : By using the local estimate recently proved by B. Choi and P. Daskalopoulos, we show that the smoothing time of convex inverse mean curvature flows is given by the smallest area of the initial tangent cone. As a corollary, convex inverse mean curvature flows on the sphere become smooth before the extinction time unless the corresponding cone splits a line. This is ongoing joint work with B. Choi.

#### May 22, 2018 Tue, 3:45–4:45PM, Eckhart 207

Bing Wang, University of Wisconsin - Madison

The extension problem of the mean curvature flow in R^3 : We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3. This is a joint work with H.Z. Li.

#### June 5, 2018 Tue, 3:00–4:00PM, Eckhart 207 Please note we will have two talks on the same day

Xinliang An, U of Toronto

How to Make a Black Hole : Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being. In this talk, with tools from hyperbolic PDE, quasilinear elliptic equations, geometric analysis and dynamical systems, we will prove that, through a nonlinear focusing effect, initially low-amplitude and diffused gravitational waves can give birth to a black hole region in our universe. This result extends the 1965 Penrose’s singularity theorem and it also proves a conjecture of Ashtekar on black-hole thermodynamics. Open problems and new directions will also be discussed.

#### June 5, 2018 Tue, 4:15–5:15PM, Eckhart 207 Please note we will have two talks on the same day

Allen Brian, USMA West Point

Using IMCF to show stability of the Positive Mass Theorem and Riemannian Penrose Inequality : In this talk we will discuss the stability of the Positive Mass Theorem (PMT) and Riemannian Penrose Inequality (RPI) for a sequence of compact regions of asymptotically flat manifolds, $U_T \subset M^3$, which are foliated by a uniformly controlled solution to Inverse Mean Curvature Flow (IMCF). The PMT result says that if the Hawking Mass of the outermost boundary of the regions, $U_T$, is approaching zero than the metric on $U_T$ is converging to the Euclidean metric in $L^2$. A similar result for the RPI will also be discussed where the metric on $U_T$ converges to the Schwarzschild metric in $L^2$. Key estimates will be introduced and ideas for how to extend this work will be explored.

### WINTER 2018

#### Jan 9, 2018 Tue, 3:45–4:45PM, Eckhart 217

Nicolau Aiex, University of British Columbia

The space of min-max hypersurfaces : We use Lusternik-Schnirelmann Theory to study the topology of the space of closed embedded minimal hypersurfaces on a manifold of dimension between 3 and 7 and positive Ricci curvature. Combined with the works of Marques-Neves we can also obtain some information on the geometry of the minimal hypersurfaces they found.

#### Jan 16, 2018 Tue, 3:45–4:45PM, Eckhart 217

Robin Neumayer, Northwestern University

The Cheeger constant of a Jordan domain without necks : n 1970, Cheeger established lower bounds on the first eigenvalue of the Laplacian on compact Riemannian manifolds in terms of a certain isoperimetric problem. The analogous problem on domains of Euclidean space has generated much interest in recent years, due in part to its connections to capillarity theory, image processing, and landslide modeling. In this talk, based on joint work with Leonardi and Saracco, we give an explicit characterization of minimizers in this isoperimetric problem for a very general class of planar domains.

#### Jan 30, 2018 Tue, 3:45–4:45PM, Eckhart 217

Kei Irie, Kyoto University

Denseness of closed geodesics on surfaces with generic Riemannian metrics : We prove that, on a closed surface with a $C^\infty$-generic Riemannian metric, the union of nonconstant closed geodesics is dense. This result follows from a more general result about periodic orbits of Reeb dynamics on contact three-manifolds. The proof uses embedded contact homology (ECH), a version of Floer homology definedfor contact three-manifolds, which was introduced by Hutchings. In particular, the key ingredient is the Weyl law'' for ECH spectral numbers, which was proved by Cristofaro-Gardiner, Hutchings, and Ramos. We also discuss a denseness of minimal hypersurfaces for genericmetrics (joint work with Marques and Neves), which was obtained by applying a similar idea to the Weyl law for volume spectrum, which was proved by Liokumovich, Marques, and Neves.

#### Feb 6, 2018 Tue, 3:45–4:45PM, Eckhart 217

Mathew Langford, The University of Tennessee

Ancient solutions of the mean curvature flow : I will present a survey of existence and rigidity results for ancient solutions of mean curvature flow. In particular, I will describe recent work (with Theodora Bourni and Giuseppe Tinaglia) on the existence and uniqueness of rotationally symmetric ancient solutions which lie in a slab. Time permitting, we will finish by describing some interesting open problems.

#### Feb 12, 2018 Mon, 3:45–4:45PM, Eckhart 202 Please note the unusual day

Theodora Bourni, The University of Tennessee

Ancient Pancakes : We show that, up to rigid motions, there is a unique compact, convex, rotationally symmetric, ancient solution of mean curvature flow that lies in a slab of width $\pi$ and in no smaller slab. This is joint work with Mat Langford and Giuseppe Tinaglia

#### Feb 20, 2018 Tue, 3:45–4:45PM, Eckhart 217

Zahra Sinaei, Northwestern University

TBA : In this talk, I discuss partial regularity of stationary solutions and minimizers u from a set \Omega\subset \R^n to a Riemannian manifold N, for the functional \int_\Omega F(x,u,|\nabla u|^2) dx. The integrand F is convex and satisfies some ellipticity, boundedness and integrability assumptions. Using the idea of quantitative stratification I show that the k-th strata of the singular set of such solutions are k-rectifiable.​​

TBA : TBA

#### Mar 6, 2018 Tue, 3:45–4:45PM, Eckhart 217

Panagiotis Gianniotis, University of Toronto

The bounded diameter conjecture for 2-convex mean curvature flow. : In this talk I address the bounded diameter conjecture for the mean curvature flow of smooth 2-convex hypersurfaces in $R^{n+1}$. In joint work with Robert Haslhofer, we prove that the intrinsic diameter of the evolving hypersurfaces is controlled, up to the first singular time, in terms of geometric information of the initial hypersurface. Moreover, this diameter estimate leads to sharp $L^{n−1}$ estimates for the curvature at each time. Our estimates extend to mean curvature flow with surgery, which allows us to obtain the optimal $L^{n−1}$ estimate for any level set flow starting from a smooth 2-convex hypersurface. This improves the $L^{n−1−\varepsilon}$ curvature estimate that was previously established in work of Head and Cheeger-Haslhofer-Naber.

### FALL 2017

#### Oct 3, 2017 Tue, 3:30–4:30PM, Eckhart 358

Antoine Song, Princeton University

Local min-max surfaces and existence of minimal Heegaard splittings : Let M be a closed oriented 3-manifold not diffeomorphic to the 3-sphere, and suppose that there is a strongly irreducible Heegaard splitting H. Previously, Rubinstein announced that either there is a minimal surface of index at most one isotopic to H or there is a non-orientable minimal surface such that the double cover with a vertical handle attached is isotopic to H. He sketched a natural outline of a proof using min-max, however some steps are non-trivially incomplete and we will explain how to justify them. The key point is a version of min-max theory producing interior minimal surfaces when the ambient manifold has minimal boundary. Some corollaries of the theorem include the existence in any RP^3 of either a minimal torus or a minimal projective plane with stable universal cover. Several consequences for metric with positive scalar curvature are also derived.

#### Oct 10, 2017 Tue, 3:30–4:30PM, Eckhart 358

Renato Bettiol, University of Pennsylvania

Non-uniqueness of conformal metrics with constant Q-curvature : The problem of finding (complete) metrics with constant Q-curvature in a prescribed conformal class is a famous fourth-order cousin of the Yamabe problem. In this talk, I will provide some background on Q-curvature and discuss how several non-uniqueness results for the Yamabe problem can be transplanted to this context. However, special emphasis will be given to multiplicity phenomena for constant Q-curvature that have no analogues for the Yamabe problem, confirming expectations raised by the lack of a maximum principle.

#### Oct 17, 2017 Tue, 3:30–4:30PM, Eckhart 358

Otis Chodosh, Princeton University

Minimal surfaces in asymptotically flat 3-manifolds : The study of minimal surfaces in asymptotically flat 3-manifolds goes back to the proof of the positive mass theorem by Schoen and Yau. I'll explain a rigidity theorem (joint with M. Eichmair) and an existence theorem (joint with D. Ketover) concerning such surfaces.

#### Oct 24, 2017 Tue, 3:30–4:30PM, Eckhart 358

Chao Li, Stanford University

A polyhedron comparison theorem in 3-manifolds with positive scalar curvature : We establish a comparison theorem for polyhedrons in 3-manifolds with nonnegative scalar curvature, answering affirmatively the dihedral rigidity conjecture by Gromov. For a large collections of polyhedrons with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less than those of the corresponding Euclidean model. We also establish the rigidity case.

#### Oct 31, 2017 Tue, 3:30–4:30PM, Eckhart 358

Robert Haslhofer, University of Toronto

Minimal two-spheres in three-spheres : We prove that any manifold diffeomorphic to S^3 and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three-manifolds. Finally, we apply our methods to solve a problem posed by S.T. Yau in 1987, and to show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp. This is joint work with Dan Ketover.

#### Nov 8, 2017 Wed, 2:00–3:00PM, Eckhart 358 There will be two seminars on this date

Hannah Alpert, The Ohio State University

Morse broken trajectories and hyperbolic volume : A large family of theorems all state that if a space is topologically complex, then the functions on that space must express that complexity, for instance by having many singularities. For the theorem in this talk, our preferred measure of topological complexity is the hyperbolic volume of a closed manifold admitting a hyperbolic metric (or more generally, the Gromov simplicial volume of any space). A Morse function on a manifold with large hyperbolic volume may still not have many critical points, but we show that there must be many flow lines connecting those few critical points. Specifically, given a closed n-dimensional manifold and a Morse-Smale function, the number of n-part broken trajectories is at least the Gromov simplicial volume. To prove this we adapt lemmas of Gromov that bound the simplicial volume of a stratified space in terms of the complexity of the stratification.

#### Nov 8, 2017 Wed, 3:00–4:00PM, Eckhart 358 There will be two seminars on this date

Jose Maria Espinar, IMPA - Brazil

Characterization of f-extremal disks : In this talk we show uniqueness for overdetermined elliptic problems defined on topological disks with regular boundary, i.e., positive solutions $u$ to $\Delta u + f(u)=0$ in $\Omega \subset (M^2,g)$ so that $u = 0$ and $\frac{\partial u}{\partial \vec\eta} = cte$ along $\partial \Omega$, $\vec\eta$ the unit outward normal along $\partial\Omega$ under the assumption of the existence of a candidate family. In particular, this gives a positive answer to the Schiffer conjecture for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains, also called {\it Serrin Problem}, in $\mathbb S ^2$. This is a joint work with L. Mazet.

#### Nov 15, 2017 Wed, 2:00–3:00PM, Eckhart 358

Gábor Székelyhidi, University of Notre Dame

New Calabi-Yau metrics on C^n : I will discuss the construction of Calabi-Yau metrics on C^n with maximal volume growth, with singular tangent cones at infinity. These generalize recent examples constructed independently by Yang Li and Conlon-Rochon

#### Nov 22, 2017 Wed, 2:00–3:00PM, Eckhart 358

Fritz Hiesmayr, University of Cambridge

Index and spectrum of minimal hypersurfaces arising from the Allen-Cahn construction : The Allen-Cahn construction is a method for constructing minimal surfaces of codimension 1 in closed manifolds. In this approach, minimal hypersurfaces arise as the weak limits of level sets of critical points of the Allen-Cahn energy functional. This talk will relate the variational properties of the Allen-Cahn energy to those of the area functional on the surface arising in the limit, under the assumption that the limit surface is two-sided. In this case, bounds for the Morse indices of the critical points lead to a bound for the Morse index of the limit minimal surface. As a corollary, minimal hypersurfaces arising from an Allen-Cahn p-parameter min-max construction have index at most p. An analogous argument also establishes a lower bound for the spectrum of the Jacobi operator of the limit surface.

### SPRING 2017

#### Apr 4, 2017 Tue, 3:00–4:00PM, Eckhart 358

Shmuel Weinberger, The University of Chicago

Persistent Homology and Gromov's theorem on closed contractible geodesics. : Gromov showed that on any closed Riemannian manifold whose fundamental group has unsolvable word problem, there are infinitely many closed nullhomotopic geodesics (of index 0). I will explain this theorem from the point of Persistent Homology of the free loop space, and then give some refinements (e.g. to less exotic fundamental groups) and extensions (to some other functionals other than energy on loops).

#### Apr 11, 2017 Tue, 3:00–4:00PM, Eckhart 358

David Wiygul, UC Irvine

The Bartnik-Bray outer mass of small spheres : In 1989 Robert Bartnik proposed a definition of quasilocal mass in general relativity. The Bartnik mass is known to enjoy several attractive properties but is not straightforward to evaluate. I will talk about a first-order estimate for a natural modification of Bartnik's definition applied to small perturbations of spheres in Euclidean space. In particular I will describe an application to the small-sphere limit in time-symmetric slices.

#### Apr 18, 2017 Tue, 3:00–4:00PM, Eckhart 358

Nick Edelen , MIT

Quantitative Reifenberg for Measures : In joint work with Aaron Naber and Daniele Valtorta, we demonstrate a quantitative structure theorem for measures in R^n under assumptions on the Jones \beta-numbers, which measure how close the support is to being contained in a subspace. Measures with this property have arisen in several interesting scenarios: in obtaining packing estimates on and rectifiability of the singular set of minimal surfaces; in characterizing L2-boundedness of Calderon-Zygmund operators; and as an "annalist's" formulation of the travelling salesman problem.

#### Apr 25, 2017 Tue, 3:00–4:00PM, Eckhart 358

Dan Ketover , Princeton University

Free boundary minimal surfaces of unbounded genus : Free boundary minimal surfaces are natural variational objects that have been studied since the 40s. In spite of this, very few explicit examples in the simplest case of the round three ball are known. I will describe how variational methods can be used to construct new examples with unbounded genus resembling a desingularization of the critical catenoid and flat disk. I will also give a new variational interpretation of the previously known examples.

#### May 02, 2017 Tue, 3:00–4:00PM, Eckhart 358

Marco Radeschi , University of Notre Dame

Minimal hypersurfaces in compact symmetric spaces : A conjecture of Marques-Neves-Schoen says that for every embedded minimal hypersurface M in a manifold of positive Ricci curvature, the first Betti number of M is bounded above linearly by the index of M. We will show that for every compact symmetric space this result holds, up to replacing the index of M with its extended index. Moreover, for special symmetric spaces, the actual conjecture holds for all metrics in a neighbourhood of the canonical one. These results are a joint work with R. Mendes.

#### May 16, 2017 Tue, 3:00–4:00PM, Eckhart 358

Pierre Albin , University of Illinois at Urbana-Champaign

Analytic torsion of manifolds with fibered cusps : Analytic torsion is a spectral invariant of the Hodge Laplacian of a manifold with a flat connection. On a closed manifold it is equal to a topological invariant known as Reidemeister torsion. I will describe joint work with Frédéric Rochon and David Sher establishing a topological expression for the analytic torsion of a manifold with fibered cusp ends (such as a locally symmetric space of rank one). We establish our result by controlling the behavior of the spectrum along a degenerating class of Riemannian metrics.

#### May 23, 2017 Tue, 3:00–4:00PM, Eckhart 358 There will be two seminars on this date

Jacob Bernstein , Johns Hopkins University

Surfaces of Low Entropy : Following Colding and Minicozzi, we consider the entropy of (hyper)-surfaces in Euclidean space. This is a numerical measure of the geometric complexity of the surface and is intimately tied to to the singularity formation of the mean curvature flow. In this talk, I will discuss several results that show that closed surfaces for which the entropy is small are simple in various senses. This is all joint work with L. Wang.

#### May 23, 2017 Tue, 4:30–5:30PM, Eckhart 358 There will be two seminars on this date

Pedro Gaspar , IMPA

Minimal hypersurfaces and the Allen-Cahn equation on closed manifolds : Since the late 70s parallels between the theory of phase transitions and critical points of the area functional have helped us to understand variational properties of certain semi-linear elliptic PDEs and spaces of hypersurfaces which minimize the area in an appropriate sense. We will discuss some recent developments in this direction which extend well-known analogies regarding minimizers to more general variational solutions. In particular, borrowing ideas from the min-max theory of minimal hypersurfaces, we study the number of solutions of the Allen-Cahn equation in a closed manifolds and solutions with least non-trivial energy.

#### May 30, 2017 Tue, 3:00–4:00PM, Eckhart 358

Or Hershkovits , Stanford University

Uniqueness of mean curvature flow with mean convex singularities : Given a smooth compact hypersurface in Euclidean space, one can show that there exists a unique smooth evolution starting from it, existing for some maximal time. But what happens after the flow becomes singular? There are several notions through which one can describe weak evolutions past singularities, with various relationship between them. One such notion is that of the level set flow. While the level set flow is almost by definition unique, it has an undesirable phenomenon called fattening: Our "weak evolution" of n-dimensional hypersurfaces may develop (and does develop in some cases) an interior in R^{n+1}. This fattening is, in many ways, the right notion of non-uniqueness for weak mean curvature. As was alluded to above, fattening can not occur as long as the flow is smooth. Thus it is reasonable to say that the source of fattening is singularities. Permitting singularities, it is very easy to show that fattening does not occur if the initial hypersurface, and thus all the evolved hypersurface, are mean convex. Thus, singularities encountered during mean convex mean curvature flow should be of the kind that does not create singularities (i.e, the local structure of the singularities should prevent fattening, without any global mean convexity assumption). To put differently, it is reasonable to conjecture that: "An evolving surface cannot fatten unless it has a singularity with no spacetime neighborhood in which the surface is mean convex". In this talk, we will phrase a concrete formulation of this conjecture, and describe its proof. This is a joint work with Brian White.

#### June 05, 2017 Tue, 3:00–4:00PM, Eckhart 358

Xin Zhou , MIT

Min-max minimal hypersurfaces with free boundary : I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the min-max method. Our result allows the min-max free boundary minimal hypersurface to be improper; nonetheless the hypersurface is still regular.

#### June 29, 2017 Thu, 3:00–4:00PM, Eckhart 202

Henrik Matthiesen , Max Plank Institute for Mathematics Bonn

Existence of metrics maximizing the first eigenvalue on closed surfaces : We show that on each closed surface of fixed topological type, orientable or non-orientable, there is a metric, smooth away from finitely many conical singularities, that maximizes the first eigenvalue among all unit area metrics. The key new ingredient are several monotonicity results relating the corresponding maximal eigenvalues. This is joint work with Anna Siffert.

### WINTER 2017

#### Jan 10, 2017 Tue, 3:00–4:00PM, Eckhart 312

Gang Liu, Northwestern University

On some recent progress of Yau's uniformization conjecture : Yau's uniformization conjecture states that a complete noncompact Kahler manifold with positive bisectional curvature is biholomorphic to the complex Euclidean space. We shall discuss some recent progress via the Gromov-Hausdorff convergence technique.

#### Jan 17, 2017 Tue, 3:00–4:00PM, Eckhart 312

John Ma, University of British Columbia

Compactness, finiteness properties of Lagrangian self-shrinkers in R^4 and Piecewise mean curvature flow. : In this talk, we discuss a compactness result on the space of compact Lagrangian self-shrinkers in R^4. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Lojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in R^4, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in R^3 by Colding and Minicozzi. This is a joint work with Jingyi Chen.

#### Jan 24, 2017 Tue, 3:00–4:00PM, Ryerson 358

Laura Schaposnik, University of Illinois at Chicago

On the geometry of branes in the moduli space of Higgs bundles. : After giving a gentle introduction to Higgs bundles, their moduli space and the associated Hitchin fibration, I shall describe how actions both on groups and on surfaces (anti-holomorphic or of finite groups) lead to families of interesting subspaces of the moduli space of Higgs bundles (the so-called branes). Finally, we shall look at correspondences between these branes that arise from Langlands duality, as well as from other relations between Lie groups.

#### Jan 31, 2017 Tue, 3:00–4:00PM, Ryerson 358

Pei-Ken Hung, Columbia University

A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold : We prove a sharp inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold. This inequality generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies on a monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established by Brendle.

#### Feb 7, 2017 Tue, 3:00–4:00PM, Ryerson 358

Lan-Hsuan Huang, University of Connecticut

Geometry of Static Asymptotically Flat 3-Manifolds : A static potential on a 3-manifold reflects the symmetry of the spacetime development and is also closely related to the scalar curvature deformation. We discuss how minimal surfaces of 3-manifold interact with the static potential and, as an application, give a new proof to the rigidity of the Riemannian positive mass theorem.

#### Feb 14, 2017 Tue, 3:00–4:00PM, Ryerson 358

Ricci flow on asymptotically Euclidean manifolds : In this talk, we prove that if an asymptotically Euclidean (AE) manifold with nonnegative scalar curvature has long time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the mass will drop to zero as time tends to infinity. Moreover, in three dimensional case, we use Ricci flow with surgery to give an independent proof of positive mass theorem.

#### Feb 21, 2017 Tue, 3:00–4:00PM, Ryerson 358

Yannick Sire, Johns Hopkins University

Bounds on eigenvalues on Riemannian surfaces : In a joint program with N. Nadirashvili, we developed some tools to prove existence of extremal metrics in conformal classes for eigenvalues of the Laplace-Beltrami operator on surfaces. I will describe these results and move to an application to the isoperimetric inequality of the third eigenvalue on the 2-sphere.

#### Feb 28, 2017 Tue, 3:00–4:00PM, Ryerson 358 There will be two seminars on this date

Henri Roesch, Duke University

Proof of a Null Penrose conjecture using a new Quasi-local Mass : We define an explicit quasi-local mass functional which is non-decreasing along all foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the null Penrose conjecture under fairly generic conditions.

#### Feb 28, 2017 Tue, 4:00–5:00PM, Eckhart 206 There will be two seminars on this date - Note the different room

Luca Spolaor, MIT

A direct approach to an epiperimetric inequality for Free-Boundary problems : Using a direct approach, we prove a 2-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce the $C^{1,\alpha}$ regularity of the free-boundary in the scalar one-phase and double-phase problems, and of the reduced free-boundary in the vectorial case, without any restriction on the sign of the component functions. In this talk I will try to explain the proof of the epiperimetric inequality in the scalar one-phase problem. This is joint work with Bozhidar Velichkov.

#### March 7, 2017 Tue, 3:00–4:00PM, Ryerson 358

Victoria Sadovskaya, Department of Mathematics - Penn State

Boundedness, compactness, and invariant norms for operator-valued cocycles.: We consider group-valued cocycles over dynamical systems with hyperbolic behavior, such as hyperbolic diffeomorphisms or subshifts of finite type. The cocycle A takes values in the group of invertible bounded linear operators on a Banach space and is Holder continuous. We consider the periodic data of A, i.e. the set of its return values along the periodic orbits in the base. We show that if the periodic data of A is bounded or contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric with respect to a Holder continuous family of norms. This is joint work with B. Kalinin.

#### Mar 14, 2017 Tue, 3:00–4:00PM, Ryerson 358

Baris Coskunuser , Boston College

Embeddedness of the solutions to the H-Plateau Problem : In this talk, we will give a generalization of Meeks and Yau's embeddedness result on the solutions of the Plateau problem to the constant mean curvature disks. arXiv:1504.00661

### FALL 2016

#### Oct 11, 2016 Tue, 3:00–4:00PM, Eckhart 207

Lu Wang, Department of Mathematics - University of Wisconsin-Madison

Asymptotic structure of self-shrinkers: We show that each end of a noncompact self-shrinker in Euclidean 3-space of finite topology is smoothly asymptotic to a regular cone or a self-shrinking round cylinder.

#### Oct 18, 2016 Tue, 3:00–4:00PM, Eckhart 207

Spencer Becker-Kahn, Department of Mathematics - MIT

Zero Sets of Smooth Functions and p-Sweepouts: It is well known that any closed set can occur as the zero set of a smooth function. But in many contexts in analysis, one does not work with arbitrary smooth functions; one works with smooth functions that vanish to only finite order. And the zero set of any such function must have some regularity and structure. With recourse to an analogy made by Gromov and some recent conjectures of Marques and Neves, I will explain the application of some general results about such zero sets to p-sweepouts (which are p-dimensional families of generalized hypersurfaces that sweepout a smooth manifold in a certain sense). In the course of doing so, I will explain a new result about the regularity of zero sets of smooth functions near a point of finite vanishing order (joint work with Tom Beck and Boris Hanin)..

#### Nov 8, 2016 Tue, 3:00–4:00PM, Eckhart 207 There will be two seminars on this date

Jonathan Zhu, Department of Mathematics - Harvard University

Entropy and self-shrinkers of the mean curvature flow: The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set.

#### Nov 8, 2016 Tue, 4:00–5:00PM, Eckhart 207 There will be two seminars on this date

Greg Chambers, Department of Mathematics - The University of Chicago

Existence of minimal hypersurfaces in non-compact manifolds: I prove that every complete non-compact manifold of finite volume admits a minimal hypersurface of finite area. This work is in collaboration with Yevgeny Liokumovich.

#### Nov 15, 2016 Tue, 3:00–4:00PM, Eckhart 207

Peter McGrath, Brown University

New doubling constructions for minimal surfaces: I will discuss recent work (with Nikolaos Kapouleas) on constructions of new embedded minimal surfaces using singular perturbation 'gluing' methods. I will discuss at length doublings of the equatorial S^2 in S^3. In contrast to earlier work of Kapouleas, the catenoidal bridges may be placed on arbitrarily many parallel circles on the base S^2. This necessitates a more detailed understanding of the linearized problem and more exacting estimates on associated linearized doubling solutions.

#### Nov 22, 2016 Tue, 3:00–4:00PM, Eckhart 207 There will be two seminars on this date

Franco Vargas Pallete, UC Berkeley

Renormalized volume: The renormalized volume V_R is a finite quantity associated to certain hyperbolic 3-manifolds of infinite volume. In this talk I'll discuss its definition and some properties for acylindrical manifolds, namely local convexity and convergence under geometric limits. If time suffices I'll also discuss some applications and further examples.

#### Nov 22, 2016 Tue, 4:00–5:00PM, Eckhart 207 There will be two seminars on this date

Christos Mantoulidis, Stanford University

Fill-ins, extensions, scalar curvature, and quasilocal mass: There is a special relationship between the Jacobi and the ambient scalar curvature operator. I'll talk about an extremal bending technique that exploits this relationship. It lets us compute the Bartnik mass of apparent horizons and disprove a form of the Hoop conjecture due to G. Gibbons. Then, I'll talk about a derived "cut-and-fill" technique that simplifies 3-manifolds of nonnegative scalar curvature. It is used in studying a priori L^1 estimates for boundary mean curvature for compact initial data sets, and in generalizing Brown-York mass. Parts of this talk reflect work done jointly with R. Schoen/P. Miao.

#### Nov 29, 2016 Tue, 3:00–4:00PM, Eckhart 207

Hung Tran, UC Irvine

Index Characterization for Free Boundary Minimal Surfaces: A FBMS in the unit Euclidean ball is a critical point of the area functional among all surfaces with boundaries in the unit sphere, the boundary of the ball. The Morse index gives the number of distinct admissible deformations which decrease the area to second order. In this talk, we explain how to compute the index from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. We also discuss applications to a conjecture about FBMS with index 4.

For further information on this seminar or this webpage, please contact Marco Guaraco at guaraco(at)math.uchicago.edu, Lucas Ambrozio at lambrozio(at)math.uchicago.edu or Rafael Montezuma at montezuma(at)math.uchicago.edu